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If the roots of the quadratic equation (b-c) x^{2} +x(c-a)+a-b=0 are equal then prove that 2b=a+c

do u have NCERT syllabus?
okk, i too have same syllabus , i will try to solve it..



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(b-c)x² + x(c-a) + (a-b) =0
To have real and equal roots, Discriminant, D =0
(c-a)² - 4(a-b)(b-c) =0
⇒c² +a² - 2ac - 4(ab -ac -b²+ bc) =0
⇒c² + a² -2ac - 4ab +4ac +4b² -4bc =0
⇒a² +4b² +c² - 4ab - 4bc +2ac =0
⇒(a -2b +c)² =0
∴(a -2b +c) =0
(a+c) =2b   (proved )
a,b and c are in A.P , where b is A.M

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